Archive of Applied Mechanics

, Volume 77, Issue 7, pp 503–521 | Cite as

Collision detection for complicated polyhedra using the fast multipole method or ray crossing

  • Beate Muth
  • Günther Of
  • Peter Eberhard
  • Olaf Steinbach
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Abstract

The purpose of this paper is a comparison of two different methods that can be used for collision detection. One method is called the ray-crossing method, a commonly used geometrical approach. The other method is the fast multipole method, usually used for boundary element methods, which is also applied for collision detection purposes here. Both methods are especially of interest when the collision for arbitrarily shaped polyhedra has to be detected. Here, both methods are described and compared for different examples of complex shaped polyhedra with up to 5 × 105 faces and more than 5 × 105 test points regarding efficiency and required calculation time.

Keywords

Contact Collision detection Fast multipole method Boundary element method Ray crossing Molecular dynamics Point-in-polygon test 
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References

  1. 1.
    Allen M.P. and Tildesley D.J. (1989). Computer Simulations of Liquids. Clarendon Press, Oxford Google Scholar
  2. 2.
    Baraff, D.: Dynamic Simulation of Non-Penetrating Rigid Bodies. Ph. D. Thesis, Technical Report 92–1275, Computer Science Department, Cornell University, Ithaka (1992)Google Scholar
  3. 3.
    Cheng H., Greengard L. and Rokhlin V. (1999). A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155: 468–498 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Eberhard P. (2000). Kontaktuntersuchungen durch hybride Mehrkörpersystem Finite Elemente Simulationen. Shaker, Aachen Google Scholar
  5. 5.
    Greengard L. (1987). The Rapid Evaluation of Potential Fields in Particle Simulation. MIT, Cambridge Google Scholar
  6. 6.
    Greengard L. and Rokhlin V. (1987). A fast algorithm for particle simulations. J. Comput. Phys. 73: 325–348 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hackbusch W. and Nowak Z.P. (1989). On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54(4): 463–491 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hobson E.W. (1955). The Theory of Spherical and Ellipsoidal Harmonics. Chelsea, New York Google Scholar
  9. 9.
    Nabors K., Korsmeyer F.T., Leighton F.T. and White J. (1994). Preconditioned, adaptive, multipole-accelerated iterative methods for three-dimensional first-kind integral equations of potential theory. SIAM J. Sci. Comput. 15: 713–735 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Newman J.N. (1986). Distributions of sources and normal dipoles over a quadrilateral panel. J. Eng. Math. 20: 113–126 CrossRefGoogle Scholar
  11. 11.
    Nishimura N. (2002). Fast multipole accelerated boundary integral equations methods. Appl. Mech. Rev. 55: 299–324 CrossRefGoogle Scholar
  12. 12.
    O’Rourke J. (1998). Computational Geometry in C, 2nd Edn. Cambridge University Press, Cambridge MATHGoogle Scholar
  13. 13.
    Of, G., Steinbach, O.: A fast multipole boundary element method for a modified hypersingular boundary integral equation. In: Efendiev, M., Wendland, W.L., (eds.) Proceedings of the International Conference on Multifield Problems. Lecture Notes in Applied Mechanics, Vol. 12, pp. 163–169. Springer, Heidelberg (2003)Google Scholar
  14. 14.
    Of G., Steinbach O. and Wendland W.L. (2005). Applications of a fast multipole Galerkin boundary element method in linear elastostatics. Comput. Vis. Sci. 8: 201–209 CrossRefMathSciNetGoogle Scholar
  15. 15.
    Of G., Steinbach O. and Wendland W.L. (2006). The fast multipole method for the symmetric boundary integral formulation. IMA J. Numer. Anal. 26: 272–296 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Perez–Jorda J.M. and Yang W. (1996). A concise redefinition of the solid spherical harmonics and its use in the fast multipole methods. J. Chem. Phys. 104: 8003–8006 CrossRefGoogle Scholar
  17. 17.
    Rapaport D.C. (1995). The Art of Molecular Dynamics Simulation. Cambridge University Press, Cambridge Google Scholar
  18. 18.
    Rjasanow, S., Steinbach, O.: The fast solution of boundary integral equations. In: Mathematical and Analytical Techniques with Applications to Engineering. Springer, Heidelberg (2007)Google Scholar
  19. 19.
    Rokhlin V. (1985). Rapid solution of integral equations of classical potential theory. J. Comput. Phys. 60: 187–207 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Sauter S.A. (2000). Variable order panel clustering. Computing 64: 223–261 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Schinner A. (1999). Fast algorithms for the simulations of polygonal particles. Granul. Matter 2(1): 35–43 CrossRefGoogle Scholar
  22. 22.
    Steinbach, O.: Numerische Näherungsverfahren für elliptische Randwertprobleme. Finite Elemente und Randelemente. B.G. Teubner, Stuttgart (2003)Google Scholar
  23. 23.
    Tausch J. (2004). The variable order fast multipole method for boundary integral equations of the second kind. Computing 72: 267–291 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Wessel, W.: Kontakterkennung räumlicher polyhedraler Körper mit Hilfe von Methoden der Molekulardynamik. Student-Thesis STUD-212, Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart (2004)Google Scholar
  25. 25.
    White C.A. and Head–Gordon M. (1994). Derivation and efficient implementation of the fast multipole method. J. Chem. Phys. 101: 6593–6605 CrossRefGoogle Scholar
  26. 26.
    Yoshida K., Nishimura N. and Kobayashi S. (1999). A fast multipole boundary integral equation method for crack problems in 3D. Eng. Anal. Bound. Elem. 23: 97–105 MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Beate Muth
    • 1
  • Günther Of
    • 2
  • Peter Eberhard
    • 1
  • Olaf Steinbach
    • 2
  1. 1.Institute of Engineering and Computational MechanicsUniversity of StuttgartStuttgartGermany
  2. 2.Institute for Computational MathematicsGraz University of TechnologyGrazAustria

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